**Euler's Combinatorial and Geometrical Mathematics**

prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

Email: malkevitch@york.cuny.edu

web page: http://york.cuny.edu/~malk/

2007 is the 300th anniversary of Leonhard Euler's birth. In honor of Euler, this course will explore two of his seminal contributions to combinatorial/geometrical mathematics (as well as some smaller gems): Euler's "traversability theorem" (a connected graph has a tour covering its edges once and only once if and only if it is even-valent) and Euler's polyhedral formula (a (convex) 3-dimensional polyhedron satisfies V + F - E = 2 (where V, F, and E, denote respectively, the number of vertices, faces, and edges of the polyhedron)). These two theorems have provided what seems to be an endless stream of exciting pure and applied results that have influenced combinatorics, topology, and geometry and that continue to inspire mathematicians and computer scientists around the world. Among the many topics to be discussed will be coloring problems for planar graphs, Steinitz's Theorem characterizing the vertex edge graphs of 3-dimensional convex polytopes, Eberhard type theorems, crossing number, and traversability problems. Many open problems will be stated.

Prerequisite:

Basic discrete mathematics.

This course will be essentially self-contained. However, due to the nature of the mathematics involved, we will be able to state a variety of old, recent, and new (to be mentioned in this course for the first time) problems in geometry and combinatorics that grow out of Euler's work.

Texts:

Unfortunately, there is no ideal text for this course. However, there are a variety of books that students may wish to consult. In addition to these I will distribute in class photocopies of several articles which are of historical interest and which survey various topics which I will treat.

Grünbaum, Branko, Convex Polytopes (Second Edition), Springer, New York, 2003. (Earlier edition: Wiley, 1967.)

Mohar, B. and C. Thomassen, Graphs on Surfaces, Johns Hopkins University Press, Baltimore, 2001.

Ziegler, G., Lectures on Polytopes, Springer-Verlag, New York, 1995. (There is also a Second Edition of this book.)